Skip to main content

Full text of "Analytical Mechanics"

See other formats


ANGULAR IMPULSE AND ANGULAR MOMENTUM    273
the angular momentum of the body due to the motion of its particles relative to the center of mass, while the second term represents the angular momentum of the body due to the motion of its particles with the center of mass. The second term depends upon the position of the center of mass relative to the axis of rotation. The first term does not at all depend upon this position. It depends upon the distribution of the particles of the body about the center of mass. The two terms are, therefore, independent; that is, if the center of mass of a body is suddenly fixed the angular momentum of the body due to the motion of its particles about the center of mass is not at all affected. On the other hand if the motion about the center of mass is destroyed the angular momentum about a given axis due to the motion of the particles of the body with the center of mass is not changed. In other words motion about the center of mass and motion with the center of mass are distinct and independent.*
As an illustration of this important fact consider two disks, Fig. 124, of equal mass, radius, and thickness, which have equal and opposite angular velocities about a common axle, and which move with the axle in a direction perpendicular to it. Suppose each of the disks to have two similarly placed holes, as shown in the figure, so that they can be made one solid piece by dropping a pin in each pair of holes when they are in line. If the rotational motion is stopped by dropping the pins into the holes, the motion of the axle goes on as if nothing had happened. On the other hand if the motion with the axle is changed or even stopped, the rotations of the disks about the axle are not at all disturbed.
* This result holds true for all bodies and systems, whether rigid or not.
u-
FIG. 124.